MR a1 Loesung Fossy

...und wie geht's weiter?

zurück zur Aufgabenstellung

Gegeben sind 7 Werte - die ersten 7 Werte. Gesucht ist eine Regel für die weiteren Werte. Nichts liegt näher, als das über ein Polynom zu lösen. $p(i)$ wobei $i$ die gewünschte Zeile ist. Wir kennen $p(0)$ ... $p(6)$.

$p(0)=1$
$p(1)=11$
$p(2)=21$
$p(3)=1211$
$p(4)=111221$
$p(5)=312211$
$p(6)=13112221$

Also gesucht ist ein Polynom, das genau das oben stehende erfüllt - sonst nix. Die Suche ist einfach, wenn man andere Polynome addiert. Ich nenne sie $q_{j}(i)$ - dieses Polynom (ich brauche 7 verschiedene solche) hat an der Stelle $j$ den Wert 1 - an den anderen (ganzzahligen) Stellen hat es den Wert 0:

$q_{j}(x)={\prod _{k\in \{0..6\}-j}{(x-k)} \over \prod _{k\in \{0..6\}-j}{(j-k)}}$

Das gesuchte $p(x)$ ist dann blos die Summe der geiegneten q's mal dem gewünschten Wert an der jeweiligen Stelle:

$p(x)=1\cdot q_{0}(x)+11\cdot q_{1}(x)+21\cdot q_{2}(x)+...+13112221\cdot q_{6}(x)$

$q_{0}(x)={{(x-1)(x-2)(x-3)(x-4)(x-5)(x-6)} \over {(0-1)(0-2)(0-3)(0-4)(0-5)(0-6)}}={{x^{6}-21x^{5}+175x^{4}-735x^{3}+1624x^{2}-1764x+720} \over 720}$

$q_{1}(x)={{(x-0)(x-2)(x-3)(x-4)(x-5)(x-6)} \over {(1-0)(1-2)(1-3)(1-4)(1-5)(1-6)}}={{x^{6}-20x^{5}+155x^{4}-580x^{3}+1044x^{2}-720x} \over -120}$

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q_2(x)={{(x-0)(x-1)(x-3)(x-4)(x-5)(x-6)}\over {(2-0)(2-1)(2-3)(2-4)(2-5)(2-6)}}={{x^6-19x^5+137x^4-461x^3+702x^2-360x}\over 48 }}

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q_3(x)={{(x-0)(x-1)(x-2)(x-4)(x-5)(x-6)}\over {(3-0)(3-1)(3-2)(3-4)(3-5)(3-6)}}={{x^6-18x^5+121x^4-372x^3+508x^2-240x}\over -36}}

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q_4(x)={{(x-0)(x-1)(x-2)(x-3)(x-5)(x-6)}\over {(4-0)(4-1)(4-2)(4-3)(4-5)(4-6)}}={{x^6-17x^5+107x^4-307x^3+396x^2-180x}\over 48}}

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q_5(x)={{(x-0)(x-1)(x-2)(x-3)(x-4)(x-6)}\over {(5-0)(5-1)(5-2)(5-3)(5-4)(5-6)}}={{x^6-16x^5+95x^4-260x^3+324x^2-144x}\over -120}}

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q_6(x)={{(x-0)(x-1)(x-2)(x-3)(x-4)(x-5)}\over{(6-0)(6-1)(6-2)(6-3)(6-4)(6-5)}}={{x^6-15x^5+85x^4-225x^3+274x^2-120x}\over 720}}

So jetzt weden nur mehr die Polynome Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q_j(x)} addiert. Das KGV der Nenner ist 720 (das führt dann zu den Faktoren 1,-6,15,-20,15,-6 und 1). Ich addiere die Faktoren vor den Potenzen von x. Das Ergebnis ist folgendes Polynom - die Koefizienten sind in der Tabelle dahinter...

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle p(x)={{a_6 x^6 + a_5 x^5 + a_4 x^4 + a_3 x^3 + a_2 x^2 + a_1 x + a_0}\over 720}}

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a_6=1\cdot 1 + 11 \cdot (-6) + 21 \cdot 15 + 1211 \cdot (-20) + 111221 \cdot 15 + 312211 \cdot (-6) + 13112221 \cdot 1 = 12.883.300}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a_5=1\cdot (-21) + 11 \cdot (-6) \cdot (-20) + 21 \cdot 15 \cdot (-19) + 1211 \cdot (-20) \cdot (-18) + 111221 \cdot 15 \cdot (-17) + 312211 \cdot (-6) \cdot (-16) + 13112221 \cdot (-15)=-194.641.140}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a_4=1\cdot 175 + 11\cdot (-6) \cdot 155 + 21\cdot 15\cdot 137 +1211\cdot (-20) \cdot 121 + 111221\cdot 15 \cdot 107 + 312211\cdot (-6) \cdot 95 + 13112221\cdot 85=1.112.190.700}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a_3=1\cdot (-735) + 11\cdot (-6) \cdot (-580) + 21\cdot 15 \cdot (-461) + 1211\cdot (-20) \cdot (-372) + 111221\cdot 15 \cdot (-307) + 312211\cdot (-6) \cdot (-260) + 13112221\cdot (-225)=-2.966.471.100}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a_2=1\cdot 1624 + 11\cdot (-6) \cdot 1044 + 21 \cdot 15 \cdot 702 + 1211\cdot (-20) \cdot 508 + 111221\cdot 15 \cdot 396 + 312211\cdot (-6) \cdot 324 + 13112221\cdot 274 =3.634.313.200}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a_1=1\cdot (-1764) + 11\cdot (-6) \cdot (-720) + 21\cdot 15 \cdot (-360) + 1211\cdot (-20) \cdot (-240) + 111221\cdot 15 \cdot (-180) + 312211\cdot (-6) \cdot (-144) + 13112221\cdot (-120)= -1.598.267.760}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a_0=720}

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a_6=}	Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 1\cdot 1}	Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle + 11 \cdot (-6)}	Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle + 21 \cdot 15}	Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle + 1211 \cdot (-20)}	Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle + 111221 \cdot 15}	Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle + 312211 \cdot (-6)}	Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle + 13112221 \cdot 1}	=12.883.300
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a_5=}	Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 1\cdot (-21)}	Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle + 11 \cdot (-6) \cdot (-20)}	Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle + 21 \cdot 15 \cdot (-19)}	Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle + 1211 \cdot (-20) \cdot (-18)}	Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle + 111221 \cdot 15 \cdot (-17)}	Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle + 312211 \cdot (-6) \cdot (-16)}	Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle + 13112221 \cdot (-15)}	=-194.641.140

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle p(x)={{12.883.300 x^6 -194.641.140 x^5 + 1.112.190.700 x^4 -2.966.471.100 x^3 + 3.634.313.200 x^2 -1.598.267.760 x + 720}\over 720}}